Multiplicative bijections of semigroups of interval-valued continuous functions

Details Multiplicative bijections of semigroups of interval-valued continuous functions Multiplicative bijections of semigroups of interval-valued continuous functions

2007-10-23

arxiv.org/abs/0710.4347v2

Mathematics

Functional Analysis

9 pages. No figures. Accepted for publication

Scientific paper

We characterize all compact and Hausdorff spaces $X$ which satisfy that for every multiplicative bijection $\phi$ on $C(X, I)$, there exist a homeomorphism $\mu : X \to X$ and a continuous map $p: X \to (0, +\infty)$ such that $$\phi (f) (x) = f(\mu (x))^{p(x)}$$ for every $f \in C(X,I)$ and $x \in X$. This allows us to disprove a conjecture of Marovt (Proc. Amer. Math. Soc. {\bf 134} (2006), 1065-1075). Some related results on other semigroups of functions are also given.

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